## The Annals of Probability

### A stochastic Burgers equation from a class of microscopic interactions

#### Abstract

We consider a class of nearest-neighbor weakly asymmetric mass conservative particle systems evolving on $\mathbb{Z}$, which includes zero-range and types of exclusion processes, starting from a perturbation of a stationary state. When the weak asymmetry is of order $O(n^{-\gamma})$ for $1/2<\gamma\leq1$, we show that the scaling limit of the fluctuation field, as seen across process characteristics, is a generalized Ornstein–Uhlenbeck process. However, at the critical weak asymmetry when $\gamma=1/2$, we show that all limit points satisfy a martingale formulation which may be interpreted in terms of a stochastic Burgers equation derived from taking the gradient of the KPZ equation. The proofs make use of a sharp “Boltzmann–Gibbs” estimate which improves on earlier bounds.

#### Article information

Source
Ann. Probab., Volume 43, Number 1 (2015), 286-338.

Dates
First available in Project Euclid: 12 November 2014

https://projecteuclid.org/euclid.aop/1415801558

Digital Object Identifier
doi:10.1214/13-AOP878

Mathematical Reviews number (MathSciNet)
MR3298474

Zentralblatt MATH identifier
1311.60069

#### Citation

Gonçalves, Patrícia; Jara, Milton; Sethuraman, Sunder. A stochastic Burgers equation from a class of microscopic interactions. Ann. Probab. 43 (2015), no. 1, 286--338. doi:10.1214/13-AOP878. https://projecteuclid.org/euclid.aop/1415801558

#### References

• [1] Amir, G., Corwin, I. and Quastel, J. (2011). Probability distribution of the free energy of the continuum directed random polymer in $1+1$ dimensions. Comm. Pure Appl. Math. 64 466–537.
• [2] Andjel, E. D. (1982). Invariant measures for the zero range processes. Ann. Probab. 10 525–547.
• [3] Assing, S. (2011). A rigorous equation for the Cole–Hopf solution of the conservative KPZ dynamics. Available at arXiv:1109.2886.
• [4] Assing, S. (2002). A pregenerator for Burgers equation forced by conservative noise. Comm. Math. Phys. 225 611–632.
• [5] Assing, S. (2007). A limit theorem for quadratic fluctuations in symmetric simple exclusion. Stochastic Process. Appl. 117 766–790.
• [6] Baik, J., Deift, P. and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119–1178.
• [7] Baik, J. and Rains, E. M. (2000). Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100 523–541.
• [8] Balázs, M., Quastel, J. and Seppäläinen, T. (2011). Fluctuation exponent of the KPZ/stochastic Burgers equation. J. Amer. Math. Soc. 24 683–708.
• [9] Balázs, M., Rassoul-Agha, F. and Seppäläinen, T. (2006). The random average process and random walk in a space–time random environment in one dimension. Comm. Math. Phys. 266 499–545.
• [10] Balázs, M. and Seppäläinen, T. (2009). Fluctuation bounds for the asymmetric simple exclusion process. ALEA Lat. Am. J. Probab. Math. Stat. 6 1–24.
• [11] Balázs, M. and Seppäläinen, T. (2010). Order of current variance and diffusivity in the asymmetric simple exclusion process. Ann. of Math. (2) 171 1237–1265.
• [12] Baxter, R. J. (1982). Exactly Solved Models in Statistical Mechanics. Academic Press, London.
• [13] Bertini, L. and Cancrini, N. (1995). The stochastic heat equation: Feynman–Kac formula and intermittence. J. Stat. Phys. 78 1377–1401.
• [14] Bertini, L. and Giacomin, G. (1997). Stochastic Burgers and KPZ equations from particle systems. Comm. Math. Phys. 183 571–607.
• [15] Borodin, A. and Corwin, I. (2012). Macdonald processes. Preprint. Available at arXiv:1111.4408.
• [16] Borodin, A., Corwin, I. and Sasamoto, T. Duality to determinants for q-TASEP and ASEP. Preprint. Available at arXiv:1207.5035.
• [17] Borodin, A., Ferrari, P. L., Prähofer, M. and Sasamoto, T. (2007). Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129 1055–1080.
• [18] Brox, T. and Rost, H. (1984). Equilibrium fluctuations of stochastic particle systems: The role of conserved quantities. Ann. Probab. 12 742–759.
• [19] Corwin, I. (2012). The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 1 1130001, 76.
• [20] Dittrich, P. and Gärtner, J. (1991). A central limit theorem for the weakly asymmetric simple exclusion process. Math. Nachr. 151 75–93.
• [21] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
• [22] Ferrari, P. A. and Fontes, L. R. G. (1994). Current fluctuations for the asymmetric simple exclusion process. Ann. Probab. 22 820–832.
• [23] Ferrari, P. A., Presutti, E. and Vares, M. E. (1988). Nonequilibrium fluctuations for a zero range process. Ann. Inst. Henri Poincaré Probab. Stat. 24 237–268.
• [24] Ferrari, P. L. and Spohn, H. (2006). Scaling limit for the space–time covariance of the stationary totally asymmetric simple exclusion process. Comm. Math. Phys. 265 1–44.
• [25] Gärtner, J. (1988). Convergence towards Burgers’ equation and propagation of chaos for weakly asymmetric exclusion processes. Stochastic Process. Appl. 27 233–260.
• [26] Gonçalves, P. (2008). Central limit theorem for a tagged particle in asymmetric simple exclusion. Stochastic Process. Appl. 118 474–502.
• [27] Gonçalves, P. and Jara, M. (2010). Universality of KPZ equation. Available at arXiv:1003.4478.
• [28] Gonçalves, P. and Jara, M. (2012). Crossover to the KPZ equation. Ann. Henri Poincaré 13 813–826.
• [29] Gonçalves, P., Landim, C. and Toninelli, C. (2009). Hydrodynamic limit for a particle system with degenerate rates. Ann. Inst. Henri Poincaré Probab. Stat. 45 887–909.
• [30] Hairer, M. (2013). Solving the KPZ equation. Ann. of Math. (2) 178 559–664.
• [31] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin.
• [32] Jara, M. D. and Landim, C. (2006). Nonequilibrium central limit theorem for a tagged particle in symmetric simple exclusion. Ann. Inst. Henri Poincaré Probab. Stat. 42 567–577.
• [33] Kardar, M., Parisi, G. and Zhang, Y. C. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 889–892.
• [34] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften 320. Springer, Berlin.
• [35] Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 1–19.
• [36] Kolmogorov, A. N. (1962). A local limit theorem for Markov chains. In Select. Transl. Math. Statist. and Probability, Vol. 2 109–129. Amer. Math. Soc., Providence, RI.
• [37] Kumar, R. (2011). Current fluctuations for independent random walks in multiple dimensions. J. Theoret. Probab. 24 1170–1195.
• [38] Landim, C., Sethuraman, S. and Varadhan, S. (1996). Spectral gap for zero-range dynamics. Ann. Probab. 24 1871–1902.
• [39] Lee, E. (2010). Distribution of a particle’s position in the ASEP with the alternating initial condition. J. Stat. Phys. 140 635–647.
• [40] Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften 276. Springer, New York.
• [41] Lu, S. L. and Yau, H.-T. (1993). Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Comm. Math. Phys. 156 399–433.
• [42] Mitoma, I. (1983). Tightness of probabilities on $C([0,1];{\mathcal{Y}}^{\prime})$ and $D([0,1];{\mathcal{Y}}^{\prime})$. Ann. Probab. 11 989–999.
• [43] Morris, B. (2006). Spectral gap for the zero range process with constant rate. Ann. Probab. 34 1645–1664.
• [44] Mueller, C. (1991). On the support of solutions to the heat equation with noise. Stochastics Stochastics Rep. 37 225–245.
• [45] Nagahata, Y. (2010). Spectral gap for zero-range processes with jump rate $g(x)=x^{\gamma}$. Stochastic Process. Appl. 120 949–958.
• [46] Prähofer, M. and Spohn, H. (2002). Current fluctuations for the totally asymmetric simple exclusion process. In In and Out of Equilibrium (Mambucaba, 2000) (V. Sidoravičius, ed.). Progress in Probability 51 185–204. Birkhäuser, Boston, MA.
• [47] Quastel, J. (1992). Diffusion of color in the simple exclusion process. Comm. Pure Appl. Math. 45 623–679.
• [48] Quastel, J. and Remenik, D. (2011). Local Brownian property of the narrow wedge solution of the KPZ equation. Electron. Commun. Probab. 16 712–719.
• [49] Quastel, J. and Valko, B. (2007). $t^{1/3}$ superdiffusivity of finite-range asymmetric exclusion processes on $\mathbb{Z}$. Comm. Math. Phys. 273 379–394.
• [50] Ravishankar, K. (1992). Fluctuations from the hydrodynamical limit for the symmetric simple exclusion in ${\mathbf{Z}}^{d}$. Stochastic Process. Appl. 42 31–37.
• [51] Rost, H. and Vares, M. E. (1985). Hydrodynamics of a one-dimensional nearest neighbor model. In Particle Systems, Random Media and Large Deviations (Brunswick, Maine, 1984). Contemp. Math. 41 329–342. Amer. Math. Soc., Providence, RI.
• [52] Sasamoto, T. and Spohn, H. (2010). One-dimensional KPZ equation: An exact solution and its universality. Phys. Rev. Lett. 104 230602.
• [53] Sasamoto, T. and Spohn, H. (2010). The crossover regime for the weakly asymmetric simple exclusion process. J. Stat. Phys. 140 209–231.
• [54] Seppäläinen, T. (2005). Second-order fluctuations and current across characteristic for a one-dimensional growth model of independent random walks. Ann. Probab. 33 759–797.
• [55] Sethuraman, S. (2001). On extremal measures for conservative particle systems. Ann. Inst. Henri Poincaré Probab. Stat. 37 139–154.
• [56] Sethuraman, S. and Xu, L. (1996). A central limit theorem for reversible exclusion and zero-range particle systems. Ann. Probab. 24 1842–1870.
• [57] Spohn, H. (1991). Large Scale Dynamics of Interacting Particles. Springer, Berlin.
• [58] Tracy, C. A. and Widom, H. (2011). Formulas and asymptotics for the asymmetric simple exclusion process. Math. Phys. Anal. Geom. 14 211–235.
• [59] van Beijeren, H., Kutner, R. and Spohn, H. (1985). Excess noise for driven diffusive systems. Phys. Rev. Lett. 54 2026–2029.
• [60] Varadhan, S. R. S. (2001). Probability Theory. Courant Lecture Notes in Mathematics 7. New York Univ. Courant Institute of Mathematical Sciences, New York.
• [61] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École D’été de Probabilités de Saint-Flour, XIV—1984. Lecture Notes in Math. 1180 265–439. Springer, Berlin.
• [62] Yin, M. (2013). A Markov chain approach to renormalization group transformations. Phys. A 392 1347–1354.